Numerical solution of variable fractional order advection-dispersion equation using Bernoulli wavelet method and new operational matrix of fractional order derivative

In this article, the Bernoulli wavelet method is used to solve the space-time variable fractional order advection-dispersion equation. The equation contains Coimbra time fractional derivatives with variable order of gamma 1(x) as well as the Riemann-Liouville space fractional derivatives with variable orders of gamma 2(x,t) and gamma 3(x,t). In fact, first, using the new operational matrices, we study the relationship between Bernoulli wavelets and piecewise functions. Then, according to the properties of piecewise functions and computing operational matrices of their fractional derivatives, we obtain operational matrices of the Bernoulli wavelet fractional derivatives. Using new operational matrices furnished from Caputo and Riemann-Liouville and also suitable collocation points, the advection-dispersion equation would be converted to a system of algebraic equations. Then, we would solve the equation numerically by utilizing a common method. Finally, the upper bound of the errors of the defined operational matrices and convergence analysis of the proposed method would be discussed. We would also reveal high accuracy of the method using some numerical samples.